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Old 2018-09-01, 19:35   #34
Bobby Jacobs
 
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In this article, they talk about rescaling gaps. There is the formula (g-T)/a. I believe that this is what I want. However, the values for maximal prime gaps are not given in the article. This article talks about primes that are r mod q. Regular prime gaps are between primes that are 0 mod 1. Therefore, a(p)=p/li(p) and T(p)=(p/li(p))(2*log(li(p))-log(p)). Then, the value is (g-((p/li(p))(2*log(li(p))-log(p))))/(p/li(p)). What are the values of this function for the known maximal prime gaps?
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Old 2018-09-01, 20:07   #35
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Regular prime gaps are between primes that are 0 mod 1.
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
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Old 2018-09-03, 03:31   #36
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Originally Posted by VBCurtis View Post
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
It has to be a typo!

The article was interesting though.
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Old 2018-09-03, 15:17   #37
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Originally Posted by VBCurtis View Post
What does 0 mod 1 mean? Or, what whole number is not 0 mod 1?
The article is about maximal gaps between primes that are r mod q. Regular maximal prime gaps are between whole numbers. Since all whole numbers are 0 mod 1, then r=0 and q=1. Therefore, regular maximal prime gaps are between primes that are 0 mod 1.
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Old 2018-09-04, 07:07   #38
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Originally Posted by Bobby Jacobs View Post
The article is about maximal gaps between primes that are r mod q. Regular maximal prime gaps are between whole numbers. Since all whole numbers are 0 mod 1, then r=0 and q=1. Therefore, regular maximal prime gaps are between primes that are 0 mod 1.
But ALL primes (except 2) are 1 Mod 2, so that is a stronger restriction. There is only 1 maximal gap that involves prime 2.
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Old 2018-09-11, 21:37   #39
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The greatest gap between primes up to n is about log2(n)-2*log(n)*log(log(n)). Therefore, a good measure would be (g-(log2(p)-2*log(p)*log(log(p))))/log(p)=(g-log2(p)+2*log(p)*log(log(p)))/log(p). Which maximal prime gap has the biggest value of (g-log2(p)+2*log(p)*log(log(p)))/log(p)?

Last fiddled with by Bobby Jacobs on 2018-09-11 at 21:40
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Old 2018-09-12, 12:37   #40
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Originally Posted by Bobby Jacobs View Post
The greatest gap between primes up to n is about log2(n)-2*log(n)*log(log(n)).
What makes you say that, and in what sense do you think that that is true?

It's not at all clear to me that we can meaningfully discuss nonleading terms when even the coefficient of the leading term is in doubt. (Some sources aren't even sure of the exponent.)
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Old 2018-10-11, 15:41   #41
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Here are the values of (g-log2(p)+2*log(p)*log(log(p)))/log(p) for the first 30 maximal prime gaps. This measure seems to have the same distribution on all maximal prime gaps.

p1:
2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009

g:
1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282

p2:
3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291

(g-log2(p1)+2*log(p1)*log(log(p1)))/log(p1):
0.01652201917, 0.9099618198, 1.441142842, 1.06365333, 0.2967400124, 1.340824126, 0.2842368037, -0.01113471657, 0.0008777107308, 1.483239756, -0.8055044796, -0.5695678433, -0.04079693315, 1.273888756, 0.1979231021, -0.1949205926, 1.015367394, 0.7377458039, -0.4610135033, 0.5222047529, 1.033263049, 0.1404118759, -0.2170613361, 1.258418936, 0.5201039801, -0.8502142355, -0.8892300766, -0.1730714975, -1.162682203, 0.2623211567

(g-log2(p2)+2*log(p2)*log(log(p2)))/log(p2):
-0.0002774068079, 0.5850019473, 1.019417059, 0.8427693921, 0.2151204219, 1.201433598, 0.2457202101, -0.03652024352, -0.02150721463, 1.44838762, -0.8100577184, -0.5731101098, -0.04431878473, 1.270502656, 0.1971322563, -0.1953011903, 1.01490609, 0.737395807, -0.4611403309, 0.5220568929, 1.033147879, 0.1403614607, -0.2170774769, 1.258402597, 0.5200965832, -0.850217021, -0.8892319752, -0.1730735353, -1.162683196, 0.2623201147

The gap with the highest value is 1327 to 1361. The gap with the lowest value is 387096133 to 387096383.

Last fiddled with by Bobby Jacobs on 2018-10-11 at 15:47
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Old 2019-02-27, 20:24   #42
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Did somebody change the title of this thread? It used to be called "A way to measure record prime gaps". Now, it is "No way to measure record prime gaps". Who vandalized the title? By the way, I have decided to call my prime gap measure (g-log2(p)+2*log(p)*log(log(p)))/log(p) the Jacobs value.
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Old 2019-02-27, 21:54   #43
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Quote:
Originally Posted by Bobby Jacobs View Post
[...]the Jacobs value.
+20 points!

If only the modern-day Inquisition would stop changing the thread titles.
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